Problem: $ F = \left[\begin{array}{rrr}2 & -1 & -1 \\ 1 & -1 & 3 \\ 2 & 0 & 0\end{array}\right]$ $ E = \left[\begin{array}{rrr}4 & 1 & 1 \\ -1 & -2 & -2 \\ 2 & 1 & 2\end{array}\right]$ Is $ F+ E$ defined?
Solution: In order for addition of two matrices to be defined, the matrices must have the same dimensions. If $ F$ is of dimension $( m \times  n)$ and $ E$ is of dimension $( p \times  q)$ , then for their sum to be defined: 1. $ m$ (number of rows in $ F$ ) must equal $ p$ (number of rows in $ E$ ) and 2. $ n$ (number of columns in $ F$ ) must equal $ q$ (number of columns in $ E$ Do $ F$ and $ E$ have the same number of rows? Yes Yes No Yes Do $ F$ and $ E$ have the same number of columns? Yes Yes No Yes Since $ F$ has the same dimensions $(3\times3)$ as $ E$ $(3\times3)$, $ F+ E$ is defined.